Download Electronic Structure of Superheavy Atoms. Revisited.

Electronic Structure of Superheavy Atoms. Revisited.
D.M. Gitman,1 A.D. Levin,1 I.V. Tyutin,2 and B.L. Voronov2
1
Institute of Physics, University of Sao Paulo, Brazil
2
Lebedev Physical Institute, Moscow, Russia
The electronic struture of an atom with Z ≤ Zc = 137 can be described by the Dirac equation
with the Coulomb field of a point charge Ze. It was believed that the Dirac equation with Z > Zc is
inconsistent and physically meaningless because the formula for the lower energy level of the Dirac
Hamiltonian formally gives imaginary eigenvalues. But a strict mathematical consideration shows
that difficulties with the electronic spectrum for Z > Zc do not arise if the Dirac Hamiltonian is
correctly defined as a self-adjoint operator, see [1]. In this article, we briefly summarize the main
physical results of that consideration in a form suitable for physicists with some additional new
details and numerical calculations of the electronic spectra.
I.
INTRODUCTION
The question of electronic structure of an atom with
large number Z of the nucleus, especially with Z that is
more than the critical value Zc = α−1 ' 137, 04, where
α is the finite structure constant, is of fundamental importance. The formulation of QED cannot be considered
really completed until an exhaustive answer to this question is given. Although nuclei with overcritical charges
can hardly be synthesized (at present, the maximum is
Z = 118), the existing heavy nuclei can imitate the supercritical Coulomb fields at collision. Nuclear forces can
hold the colliding nuclei together for 10−19 s or more.
This time is enough to effectively reproduce the experimental situation where the electron experiences the supercritical Coulomb field [2]. The electronic structure of
an atom with Z ≤ Zc can be described by the Dirac equation, which gives relativistic electronic spectra in agreement with experiment [3]. For such Z a complete set of
solutions of the Dirac equation exists, and a relativistic
quantum mechanics of an electron in such a Coulomb
field can be constructed. As for the Dirac equation with
the Coulomb field with Z > Zc , it was considered inconsistent and physically meaningless [4–6]. One of the
standard arguments is that the formula for the lower energy level,
q
2
2
(1)
E = mc 1 − (Zα) ,
formally gives imaginary result for Z > Zc . This difficulty of the imaginary spectrum was attributed to an inadmissible singularity of the supercritical Coulomb field
at the origin for a relativistic electron, see [7]. It was believed that this difficulty can be eliminated if a nucleus
of some finite radius R is considered. It was shown that
with cutting off the Coulomb potential with Z < 173 at
a radius R ∼ 1, 2 × 10−12 cm, the Dirac Hamiltonian has
physically meaningful spectrum and eigenstates [8, 9].
But even in the presence of the cutoff, another difficulty
arises at Z ∼ 173. Namely, the lower bound state energy descends to the upper boundary E = −mc2 of the
lower continuum, and it is generally agreed that in such a
situation, the problem can no longer be considered a oneparticle one because of the electron-positron pair produc-
tion, which, in particular, results in a screening of the
Coulomb potential of the nucleus.
Not disputing the fact that taking account of a finite
size of the nucleus corresponds to a more realistic setting
up the problem, we do not agree with the assertion that
the Dirac Hamiltonian with the Coulomb field of overcritical point-like nucleus charge is inconsistent. The abovementioned difficulties with the spectrum for Z > Zc do
not arise if the Dirac Hamiltonian is correctly defined as
a self-adjoint (s.a.) operator. A rigorous mathematical
treatment of all the aspects of this problem including a
spectral analysis of the Hamiltonian based on the theory of s.a. extensions of symmetric operators and the
Krein method of guiding functionals was presented in
[1, 10, 11]. It was demonstrated that from a mathematical standpoint, a definition of the Dirac Hamiltonian as a
s.a. operator presents no problem for arbitrary Z. A specific feature of the overcritical charges is a nonuniqueness
of the s.a. Dirac Hamiltonian, but this
√ non
uniqueness is
characteristic even for Z > Zs =
3/2 α−1 ' 118, 68.
For each Z ≥ Zs , there exist a family of s.a. Dirac Hamiltonians parametrized by a finite number of extra parameters (and specified by additional boundary conditions at
the origin). The existence of these parameters is a manifestation of a nontrivial physics inside the nucleus. A real
spectrum and a complete set of eigenstates can be evaluated for each Hamiltonian, so that a relativistic quantum
mechanics for an electron in such a Coulomb field can be
constructed. In the present article, we briefly summarize
all the previously obtained formal results in a form more
suitable for physicists with some additional details and
numerical calculations of the electronic spectra.
II.
DIRAC HAMILTONIAN WITH COULOMB
FIELD
We consider an electron of charge −e < 0 and mass
m moving in the Coulomb field of a charge Ze > 0. We
describe this field by a scalar electromagnetic potential
of the form A0 = Zer−1 , we set ~ = c = 1 in what follows. A behavior of the electron in the Coulomb field
is governed by the Dirac Hamiltonian Ĥ (Z) that is a
2
s.a. operator in the Hilbert space of square-integrable
bispinors Ψ (r) . On its domain Ĥ (Z) acts by the differential operation
Ȟ (Z) = γ 0 (γp̌ + m)−qr−1 , p̌ = −i∇, r = |r| , q = Zα.
In the problem under consideration there are three com2
muting s.a. operators Jˆ , Jˆz , and K̂, where Jˆ the total
angular momentum, and K̂ a spin operator,
h
i
Jˆ = L̂ + Σ/2, L̂ = [r × p̂] , K̂ = γ 0 1 + ΣL̂ .
All they commute with Ĥ
P(Z). Any bispinor Ψ (r) can be
represented as Ψ(r) =
j,M,ζ Ψj,M,ζ (r), where Ψj,M,ζ
are bispinors of the form
1 Ωj,M,ζ (θ, ϕ)f (r)
,
Ψj,M,ζ (r) =
r iΩj,M,−ζ (θ, ϕ)g (r)
Ωj,M,ζ are spherical spinors, f (r) and g (r) are radial
functions, and j = 1/2, 3/2, ..., M = −j, −j + 1, ..., j,
2
ζ = ±1. Bispinors Ψj,M,ζ are eigenvectors of Jˆ , Jˆz , and
K̂,
2
Jˆ Ψ = j(j + 1)Ψ, Jˆz Ψ = M Ψ, K̂Ψ = −ζ(j + 1/2)Ψ .
The stationary Schrödinger equation Ĥ (Z) Ψ (r) =
EΨ (r) is reduced to a radial equation ĥ (Z, j, ζ) F (r) =
EF (r), where F are doublets (with components f (r) and
g (r)) from the Hilbert space L2 (R+ ) = L2 (R+ )⊕L2 (R+ )
and ĥ (Z, j, ζ) are some s.a. radial Hamiltonians acting
on the doublets F (r) by radial differential operations
ȟ (Z, j, ζ) = −iσ2 dr +ζ(j +1/2)r−1 σ1 −qr−1 +mσ3 . (2)
On basis of the differential operations ȟ (Z, j, ζ) one
can construct all possible corresponding s.a. radial
Hamiltonians ĥ (Z, j, ζ) using von Neumann theory of
s.a. extensions of symmetric operators. All the s.a.
operators ĥ (Z, j, ζ) are defined by their definition domains Dĥ(Z,j,ζ) where they act by the same differential operations (2). By the way of construction, all
these s.a. operators ĥ (Z, j, ζ) represent s.a. extensions
of the corresponding so-called initial symmetric operators ĥin (Z, j, ζ) with the definition domains Dĥin (Z,j,ζ) =
D(R+ )⊕D(R+ ), where D(R+ ) is a space of smooth functions f (r) with a compact support on the interval R+ .
At the same time, all these s.a. operators ĥ (Z, j, ζ)
represent s.a. restrictions of the operators ĥ+
in (Z, j, ζ)
∗
that act on the so-called natural domains Dȟ(Z,j,ζ)
(R+ )
of doublets F ∈ L2 (R+ ) that are absolutely continuos
in R+ and, in addition, ȟ (Z, j, ζ) F ∈ L2 (R+ ). Thus,
∗
Dĥ+ (Z,j,ζ) = Dȟ(Z,j,ζ)
(R+ ). The above mentioned rein
strictions are given by some s.a. boundary conditions
∗
such that Dĥ(Z,j,ζ) ⊆ Dȟ(Z,j,ζ)
(R+ ). Just after all
ĥ (Z, j, ζ) are defined as s.a. operators a corresponding s.a. Dirac Hamiltonian can be restored in a unique
way. A result of constructing s.a. radial Hamiltonians
ĥ (Z, j, ζ) essentially depends on the values of the parameters Z and j. There are two regions of these parameters in the semiplane j, Z, we call them nonsingular
and singular ones, where the problem of s.a. extensions
has principally different solutions. These regions are separated by the singular curve Z = Zs (j), where
Zs (j) =
p
j (j + 1)α−1
such that the nonsingular and singular regions are defined
by the respective inequalities Z ≤ Zs (j) and Z > Zs (j).
The values Zs (j) = 118, 68; 265, 37; ... , can be called
the singular Z-value for a given j. Below, we consider
s.a. radial Hamiltonians ĥ (Z, j, ζ) and their spectra in
the nonsingular and singular regions separately. Here we
list common for both regions properties of these Hamiltonians. It turns out that the spectrum of each s.a. Hamiltonian ĥ (Z, j, ζ) is simple (nondegenerate) and contains
a continuous part occupying the two semiaxis E ≤ −m
and E ≥ m and a discrete part {Enζ (Z, j, ζ)} located
in the interval |E| ≤ m. The quantum numbers nζ take
the values: n1 = 1, 2, ...; n−1 = 0, 1, 2, .... The discrete
spectrum is always accumulated at the point E = m,
and asymptotic form of the difference Enζ (Z, j, ζ) − m as
nζ = n → ∞ is given by the well-known non-relativistic
−1
. Eigenfunctions of the
formula Ennonrel = −mq 2 2n2
discrete spectrum and generalized eigenfunctions of the
continuous spectrum form a complete orthonormalized
system in L2 (R+ ).
III.
NONSINGULAR REGION
In the nonsingular region, Z ≤ Zs (j), deficiency indices of each operator ĥin (Z, j, ζ) are zero and the operator ĥ (Z, j, ζ) = ĥ+
in (Z, j, ζ) is a unique s.a. extension of ĥin (Z, j, ζ) with the definition domain Dĥ(Z,j,ζ) =
∗
∗
(R+ ). The functions belonging to Dȟ(Z,j,ζ)
(R+ )
Dȟ(Z,j,ζ)
have the following asymptotic behavior
F (r) = O(r1/2 ), r → 0; F (r) → 0, r → ∞.
A discrete spectrum {Enζ (Z, j, ζ)} of each Hamiltonian
ĥ (Z, j, ζ) has the form
Enζ (Z, j, ζ) = p
m (nζ + γ)
, γ=
q 2 + (nζ + γ)2
q
2
(j + 1/2) − q 2 ,
(3)
which is the well-known Sommerfeld spectrum of the
Dirac electron in the Coulomb field with Z in the nonsingular region. This result justifies the standard formal
treatment of the Dirac Hamiltonian with Z in the nonsingular region in the physical literature where the Dirac
Hamiltonian is identified with the differential operation
ȞD (Z) and the natural domain is implicitly assumed.
3
IV.
SINGULAR REGION
In the singular regions, Z > Zs (j), the deficiency indices of the operator ĥin (Z, j, ζ) are (1, 1), and therefore, there exists a family {ĥν (Z, j, ζ)} of s.a. extensions of ĥin (Z, j, ζ) parametrized by a parameter ν ∈
[ − π/2, π/2], −π/2 ∼ π/2. At the same time, each
ĥν (Z, j, ζ) is a nontrivial restriction of ĥ+
in (Z, j, ζ), such
∗
(R+ ). The position of the disthat Dĥν (Z,j,ζ) ⊂ Dȟ(q,κ)
(ν)
crete energy levels Enζ (Z, j, ζ) essentially depends on ν,
in particular, there exists a value ν = ν−m , for which the
lower energy level coincides with the boundary E = −m
of the lower continuous spectrum.
Technically, it is convenient to divide the singular region into three subregions, we call them subcritical, critical, and overcritical regions. The subregions are distinguished by a character of asymptotic boundary conditions at the origin specifying the domains Dĥν (Z,j,ζ) of the
operators ĥν (Z, j, ζ) and providing their self-adjointness.
The boundary conditions are similar in each subregion,
which provides similar solutions of the corresponding
spectral problems. In what follows, we describe these
subregions, the domains Dĥν (Z,j,ζ) in these subregions,
and some details of discrete spectra.
A.
Subcritical regions
The subcritical region is defined by the inequality
Zs (j) < Z ≤ Zc (j) , where
Zc (j) = (j + 1/2) α−1 = 137, 04; 274, 08; ... .
(4)
In the subcritical region, the s.a. radial Hamiltonians
ĥν (Z, j, ζ) are specified by s.a. boundary conditions,
(ν)
FIG. 1. ν-dependence of energy levels Enζ (121, 1/2, ζ = ±1)
and Z-dependence of ν−m , j = 1/2.
B.
Critical region
The critical region is the critical curve Z = Zc (j). For
integer Z, this region does not exist if the finite structure
constant α is an irrational number, see (4). In particular,
this region certainly is absent for j = 1/2. In the critical
region, the s.a. radial Hamiltonian ĥν (Z, j, ζ) is specified
by s.a. boundary conditions at the origin of the form
F (r) = c [d0 (r) cos ν + d+ sin ν] + O(r1/2 ln r), r → 0,
where d0 (r) are some doublet with the asymptotic behavior d0 (r) = O (ln mr) as r → 0. Here the discrete
(ν)
spectrum {Enζ (Z, j, ζ)} is determined by the equation
f (E) cos ν − sin ν
= 0, f (E) = ln(2τ /m)
F (r) = c[(mr)γ d+ cos ν+(mr)−γ d− sin ν]+O(r1/2 ), r → 0,
f (E) sin ν + cos ν
q
ζ − (τ + ζm) /E
2
where 0 < γ = (j + 1/2) − q 2 < 1/2, c is an arbitrary
+ ψ(− (j + 1/2) E/τ ) +
− 2ψ(1),
2 (j + 1/2)
complex number, and d± some constant doublets. The
(ν)
discrete spectrum consists of the points {Enζ (Z, j, ζ)}
where ψ(x) = Γ0 (x)Γ−1 (x). The spectrum can be explicthat obey the equation
itly evaluated in the case of ν = ±π/2, where we have
ω(E) cos ν + Γ(1 − 2γ) sin ν
mnζ
= 0, ω (E)
(Z, j, ζ) = q
En(±π/2)
,
(5)
ω(E) sin ν − Γ(1 − 2γ) cos ν
ζ
2
2
(j
+
1/2)
+
n
−1
ζ
Γ(1 + 2γ)Γ(−γ − qEτ )[q(m − E) − (κ + γ)τ ]
=
,
2γ
Γ(γ − qE/τ )[q(m − E) − (κ − γ)τ ](2τ /m)
which is eq. (3) with γ = 0.
√
where κ = ζ(j +1/2), and τ = m2 − E 2 . The spectrum
can be evaluated explicitly for the cases of ν = ±π/2 and
ν = 0. For ν = ±π/2, we have
C. Overcritical region
(nζ − γ)m
(Z, j, ζ) = p
En(±π/2)
.
ζ
q 2 + (nζ − γ)2
(0)
For ν = 0, the spectrum Enζ (Z, j, ζ) is given by eq.
(3), of course, with new values of γ. Thus, the known
textbook results (in particular the Sommerfeld spectrum)
correspond to ν = 0.
The overcritical region is defined by the inequality
Z > Zc (j). In this region, the s.a. radial Hamiltonians ĥν (Z, j, ζ) are specified by s.a. boundary conditions
at the origin of the form
F (r) = c eiν (mr)iσ ρ+ + e−iν (mr)−iσ ρ− +O(r1/2 ), r → 0,
4
(ν)
FIG. 3. ν-dependence of energy levels Enζ (180, 1/2, ζ = ±1).
(ν)
FIG. 2. ν-dependence of energy levels Enζ (138, 1/2, ζ = ±1)
and Z-dependence of ν−m , j = 1/2.
q
2
where σ = q 2 − (j + 1/2) > 0, c is an arbitrary complex number, and ρ± some constant doublets. Its discrete
(ν)
spectrum {Enζ (Z, j, ζ)}is determined by the equation
#
3
2τ
1 X
∗
[ln (Ba ) − ln (Ba )] + σ ln
− ν = 0,
cos
2i a=1
m
"
where B1 = −2iσ, B2 (E) = iσ − Eqτ −1 , and B3 (E) =
τ (j + 1/2 − iζσ) − ζq(m − E).
V.
CONCLUDING REMARKS
The total s.a. Dirac Hamiltonian Ĥ (Z) with Z ≤
118 is defined uniquely.For Z ≥ 119, there is a family
{Ĥν1 ,...,ν∆ (Z)} of possible total s.a. Dirac Hamiltonians.
[1] B.L. Voronov, D.M. Gitman, and I.V. Tyutin, Theor.
Math. Phys. 150(1) (2007) 34
[2] W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer-Verlag, Berlin 1985)
[3] H.A. Bethe and E.E. Salpeter, Quantum Mechanics
of One- and Two-Electron Systems, Encyclopedia of
Physics, Vol. XXXV/1 (Julius Springer, Berlin 1957)
[4] P.A.M. Dirac, Proc. Roy. Soc. Lond., A 117 (1928) 610;
A 118 (1928) 351; C.G. Darwin, Proc. Roy. Soc. Lond.,
A 118 (1928) 654; W. Gordon, Zs. Phys. 48 (1928) 11
[5] M.E. Rose, Relativistic Electron Theory (Wiley, New
York 1961)
[6] A.I. Akhiezer and V.B. Berestetskiǐ, Quantum Electrodynamics (Interscience Publishers, New York 1965)
The family is parametrized by the parameters νi ∈
[ − π/2, π/2], −π/2 ∼ π/2, i = 1, ..., ∆. The number
∆ of the parameters is given by ∆ = 2k(Z), where the
integer k(Z) = (1/4 + Z 2 α2 )1/2 − δ, 0 < δ ≤ 1. Any specific s.a. Dirac Hamiltonian Ĥν1 ,...,ν∆ (Z) corresponds to
a certain prescription for a behavior of an electron at the
origin. The general theory thus describes all the possibilities that can be offered to a physicist for his choice. This
choice is a completely physical problem. We believe that
each s.a. Dirac Hamiltonian with superstrong Coulomb
field can be understood through an appropriate regularization of the potential and a subsequent limit process
of removing the regularization. We recall that a physical
interest in the electronic structure of superheavy atoms
was mainly motivated by a possible pair creation in the
superstrong Coulomb field. Consideration of this effect in
the framework of the most simplest model of a point-like
nucleus was accepted to be impossible due to the conclusion (which is wrong as it is clear now) that this model
is mathematically inconsistent [7]. We believe that the
described rehabilitation of the model allows returning to
a consideration of the particle creation in this model that
provide great scope for analytical studies.
[7] Ya. B. Zel’dovich and V.S. Popov, Soviet. Physics Uspekhi, 14 (1972) 673
[8] I. Pomeranchuk, Ya. Smorodinsky, Journal of Physics
(USSR) 9 (1945) 97; S.S. Gershtein, Ya.B. Zel’dovich,
Soviet Phys.-JETP 30 (1970) 358
[9] V.S. Popov, Soviet Nucl. Phys. 12 (1970) 429; 14 (1071)
458; Soviet Phys.-JETP 60 (1971) 1228
[10] B.L. Voronov, D.M. Gitman, I.V.Tyutin, Russian
Physics Journal, 50/1 (2007) 1; 50/9 (2007) 853; 51/2
(2008) 115
[11] D.M. Gitman, I.V.Tyutin, B.L. Voronov, Self-adjoint Extensions in Quantum Mechanics. General theory and applications to Schrödinger and Dirac equations with singular potentials. (Birkhäuser, Boston 2012) in press